Gap Theorems Of Hypersurface With Constant Mean Curvature In A Unit Sphere

In this paper, we mainly study the gap of the second fundamental form of hypersurface with constant mean curvature in the unit sphere.There have been many good results when M is minimal sub manifold. In the first section we will introduce the context and make a general description on the recent researches. In the second we will introduce the complexion of the development for this problem. In the fourth we'll discuss the second gap on special conditions using similar methods when the mean curvature is constant (maybe notzero), and find the second gap (?). Here when H=0,we haveδ=2/3. And a false claim was given in [13], we will cite his sketch of theproof pointing out his mistake. In the last section we'll also explore the third gap(?). Whereas theform of the third gap is particularly complex, we'll improve it in this section, andget the more sententious value (?) under the additional conditions.We'll give some definitions and formulas in the third section to prove the theorems.